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no. 156-163 
c o p , 2 

Digitized by the Internet Archive 

in 2013 

no. 159 
cop. Z 





Lo van Biljon 

January 20^ 196^- 

This work was supported in part "by the 
Office of Naval Research under Contract No, Nonr-l834(l5 ) 


List of Symbols ........ ............. 

Summary .............................. v 

1. INTRODUCTION ......................... 1 

2. FASTEST POSSIBLE SWITCHING .................. 4 

2.1 Introduction .......... ............ . 4 

2.2 Outline of Approach ......... 4 

2.3 Junction Charge Transfer ................. 5 

2.3.1 Accelerating Force ..... ..... 6 

2.3.2 Mobility 6 Influence of High Field ......... 7 Average Carrier Velocity ......... 8 Energy Transfer from Lattice ....... 8 

2.3*2.4 Mobility Due to Lattice Scattering .... 11 Mobility Due to Impurity Coulomb Fields . 13 

2.4 Transit Time ....................... 14 

2.4.1 Voltage Influence on Transit Time ......... 15 


3.1 Electron Current in Depletion Region ........... 20 

4. THERMAL INFLUENCE ............ .......... . 23 

5o MICROPLASMA MODEL ....................... 25 

5.1 Observed Behavior .................... 25 

5.1.1 Current-Voltage Relation ............. 25 

5.2 Electrical Equivalent Circuit .............. 29 

5.3 Energy Considerations .................. 36 

5.3.1 Determination of dg/cit .............. 38 

5„4 Energy Gained ...................... 42 

5»5 Energy Lost to Lattice .................. 44 

5.5.1 Variation of T-, with Time ............. 44 

5.6 Complete Expression for de/cH .............. 46 

6. MICROPLASMA CHARACTERISTICS . . ..... . . . . . .... . 50 

6.1 General Behavior ..................... 50 

6.2 Experimental Set-Up ................... 50 

6.3 First Experimental Results „ . . ... .... . . . . . . 52 

6 oh Attempts at Using Sampling Techniques , . . „ . . . . . „ 56 

6.5 Delay in Switch-On after Simulation ......... . 58 

7. CONCLUSION ....... .......... ........ . 60 

APPENDIX ............................. 62 

REFERENCES ............................. 64 



A, B, C, K constants 

C capacitance 

D diffusion constant 

E electric field strength 

E_ initial field in depletion region 

E critical field required for breakdown 

E kinetic energy of carrier 

F force 

I electric current 

I„ final current of a breakdown region 

J current density 

L diffusion length 

M multiplication factor 

N carrier density 

N_ donor concentration 

N. acceptor concentration 

P rate of power dissipation 

Q electric charge 

R resistance 

T temperature 

T initial temperature 

T, lattice temperature 

T^ carrier effective temperature 

V voltage 

W depletion layer width 

e base of Naperian log 

k Boltzmann constant 

k' heat conductivity of silicon 

I diameter of microplasma discharge 

m mass of carrier 

n carrier concentration; integer 

q electronic charge 



t, At time 

u phonon velocity in silicon at T 

v carrier velocity on transport 

x linear dimension 

a. ionization coefficient 


e energy 

e . ionization energy 

u mobility 

u. impurity scattering mobility 

u low field mobility 

T carrier transit time; time constant 



The transit time of electrons across a p-n junction in which an 
electric field of 10 v/m is present is investigated and proven to be about 
1 picrosecond. It is shown that carrier distribution adds about 25 per cent 
to this time when a pulse is considered and that increasing the applied voltage 
does not necessarily decrease the transit time, 

The energy exchange between electric field, carriers, and lattice is 
investigated and a cause of random pulse formation suggested from the results 

Experimental results are presented, obtained on Si junctions in which 
breakdowns occur. It is suggested that small breakdowns are established at 
speeds too fast for conventional oscilloscopes of sufficient sensitivity. It 
is proposed that the experimental technique of photon stimulation used here 
be further refined to allow the display of the true waveshape in time of micro- 
plasma breakdown. 



The application of p-n junction devices in fast switching and 
amplifying circuits has met with great success in many areas of electronics. 
However, the extreme speed of operation required of modern equipment has for 
some time already markedly influenced the design and fabrication of special 
semiconductor units in an endeavor to increase their speed , By these require- 
ment, transistors and diodes have been taken into the nanosecond region and at 
the present time, this order of magnitude seems the limit attainable with 
existing devices. 

A pulse rise time of one or more nanoseconds is considered fast, and 
a fall time of this duration is relatively difficult to obtain with transistors. 

Compromising depletion layer capacitance and spreading resistance for 
a junction still capable of practical currents without excessive temperature 
rise, produces junction constants severely limiting the speed of response,, A 
capacitance of 2 to 5 pf with an effective series resistance of between 20 and 
200 ohms is becoming common so that the basic RC rise time is of the order of a 
nanosecond or two. Some time ago avalanche circuits of special design proved 
capable of fractional nanosecond rise time but the fall time was about an order 
or more longer. With ever better junctions becoming available, avalanche break- 
down seems to hold much promise of doing considerably better than has been 
achieved up to now. The special response characteristics of "hard" breakdown 
junctions hold promise of successful application in picosecond circuits, possibly 
proving a breakthrough as far as junction speed is concerned. 

The salient features of this special "plasma" type of breakdown 
(being termed either micro- or macroplasma rather arbitrarily^ depending upon 
the cross-sectiOn of the breakdown) are the following: 

Extreme speed of current build-up - " J (10 sec). 

Practically no junction voltage change, thus limiting charging 

time for depletion layer capacitances <, 

5 -11 
Extreme speed of current switch-off (10 sec). 

Apart from the desirable characteristics mentioned above the following 
attributes of this type of breakdown greatly enhance its usefulness in all types 
of electronic circuitry; 


Square , amplitude limited pulses are produced (see Fig. l). 

The average "on" or "off" ratio in the pulse train is smoothly- 
variable from zero to infinite (see Fig. 10). 

The pulse formation and extinction is accompanied by photon 

Pulse formation is directly influenced by the presence of 

photons or other ionizing agents. 

This report deals with some aspects of this type of junction 
breakdown as observed in silicon structures. The diodes used in this 
investigation were kindly donated by the Shockley Transistor Laboratories of 
the Clevite Company,, Palo Alto^, California. 


Current pulse with voltage just 
above breakdown value. 

Current pulse with voltage about 
50 mv above breakdown value . 


10 microsec/cm 
20 microamp/cm 

Vertical : 

10 microsec/cm 
20 microamp/cm 



2A63 Diff. Amp. 
(300 Kc/sec pass .) 
band) .in 56IA 

Figure 1. Typical Breakdown Current Pulses in Si Junction 



2.1 Introduction 

It is proposed to analyze the fastest possible response obtainable 
with a p-n junction. This entails firstly a study of the speed of transfer of 
charge carriers across a junction as well as the shape of a charge packet 
traversing the depletion region „ 

The phenomena of microplasma formation is coupled to a first-order 
analysis of thermal effects in the junction while in a separate section due 
consideration is given to RC time constant effects. 

2.2 Outline of Approach 

Conventional p-n junctions used in high-frequency applications have 
a depletion layer capacitance of the order of 5 pf • Many factors militate 
against the reduction of this figure without sacrifice of speed. Lowering the 
capacitance by having a wider depletion width requires higher resistivity 
material to be used, thus increasing series spreading resistance. Also, 
decreasing the junction area poses severe technological difficulties as well 
as imposing current limitations due to thermal effects. 

A present-day compromise indicates a junction depletion layer 
capacitance of about 5 pf and a spreading resistance of about 50 ohms. An RC 
time of about one-half nanosecond thus results meaning that it will be dif- 
ficult to adequately pass pulses of around one or two nanosecond duration. 

It seems at present that the only way of utilizing existing junctions 
at speeds faster than those mentioned above is to have them operate in the so- 
called "microplasma" breakdown region. 

The phenomenon of microplasma formation has been observed by many 

f> 7 fi 
workers and several models for this type of operation have been proposed. ' ' 

The definite advantages of this mode of operation are the following: 

The effective, active area of the junction is greatly reduced; 
typical microplasmas have diameters of 2 or 3 micron. 

While going from "on" to "off" and vice versa, the voltage 
across the discharge region hardly changes; this eliminates 


much of the need for transport of charges to and from the 
effective junction depletion layer capacitance. 

An avalanche type of breakdown characteristic provides 

extremely fast current build-up; estimated rise times are of 

the order of 10 seconds (see paragraph 2.k). 

At present the most severe disadvantage of this operating mode is that junctions 

exhibiting this phenomenon reliably are not readily available. Recent develop- 


ments though indicate that this is a temporary situation. The long-standing 

objection to avalanche operation, that of thermal run-away, is also fast being 
eliminated by the production of near-perfect low saturation current junctions. 
Silicon junctions have been operated in the microplasma mode for several hours 
without noticeable tendency towards runaway or change in characteristics. 

In this report an evaluation is made of the more important factors 
influencing microplasma behavior of p-n junctions. 

2.3 Junction Charge Transfer 

The charge transfer to be considered here differs from conventional 
charged particles acted upon by an electric field in that due consideration has 
to be given to the fact that such intense electric fields are present that the 
mobility is affected. 

Consider the simplest possible model--a depletion region of width W_, 
in series with a pure voltage source V, and no extra resistance in the circuit. 




Figure 2. Simple Model Considered 


Consider as a starting point the transfer of a single charged particle across 
the depletion region. 

The velocity of transport will depend upon three main factors^ viz.: 

Accelerating force 

Mean free path 


These factors have to be briefly considered more closely. 

2„3«1 Accelerating Force 

Force = Charge x Field Strength 
If each discrete charge is equal to q, the force will be F, where: 

F = q • E (1) 


E - (V/W) volts/meter 

2o3«2 Mobility 

In its simplest form the mobility is expressed as the velocity 
resulting from unit field strength, i,e, ; 

u = v/e (meters/sec) per (volt/meter) 

In a first-order approach one could find the mobility by stating" 

q ' E ~ m ° x 

and by assuming a mean free time % 3 the distance travelled in this time may be 
found by twice integrating the above to give : 


<£ 2 
X = 2m" T 

and since 


v = — 

~ T 

the mobility follows as : 

H=— = ^ (2) 

^ TE 2m K ' 

Now, equation (2), denoting an average mobility is perfectly in order while 

the field strength in the depletion region is relatively low--of the order of 

10 volts/meter. 

At higher fields, however, the energy gained by the carriers from the 
field during a mean free time, may be a significant fraction of the mean carrier 
energy (i.e., 3/2 kT) so that the lattice and the carriers are not in thermal 
equilibrium any more. It thus becomes necessary to consider the mobility as a 
function of the field strength in the depletion region. Influence of High Field 

The field strength influences the mobility mainly through influence 
of the mean velocity. 

As before, the mobility may be defined by 

^ = E 

but the average velocity v needs to be considered further, 


2.3»2.2 Average Carrier Velocity 

Carriers obtain their energy mainly in two ways: 

Direct interaction with the electric field. 

Scattering by the lattice and by impurity centers . 

Interaction with the electric field was mentioned in paragraph 2„3<>1 so that 
now the two scattering mechanisms have to be evaluated. 

At room temperature, impurity scattering is highly elastic and thus 
not important for energy transfer considerations. However, in present-day 
structures, it has been observed (see paragraph 6„3) that better microplasma 
response is obtained at low ambient temperatures (e.g.; that of liquid nitrogen' 
so that in a more complete analysis elastic scattering may not be assumed. 

2,3«2„3 Energy Transfer from Lattice 

Let attention be confined to electrons as the mobile carriers in the 
depletion region. 

It may be shown that in a crystal at temperature T, (i,e,, lattice 

temperature T ), the average energy gain de, upon scattering is given by: 

de = kmn 

1 - 


21sH j 



m = mass of electron 

u = velocity of sound in crystal 

k = Boltzmann's constant 

T$ = K.E, of particle prior to collision with lattice 

However, where (3) gives the average energy gain, it now becomes necessary also 
to use the concept of "effective" temperature and velocity ; ■ :-... 


A reasonable assumption at the outset is that the particles in the 
depletion region have a Maxwellian velocity distribution. ' For a total 
number of carriers N, the number having velocities between v and (v + dv) will 
be given by: 


UN 2 a 2 

v e • dv 

3 ^7 ~ 


where a is some constant . 

Assigning to a the value indicated by Smith ( loc . cit . page 159) the 
complete expression for the number of carriers in the velocity interval dv, 
will be: 



2kT 2 2 

N(v)dv = ^— - e d • v dv (h) 

(o~w ^3/2 ~ ~ 

(2«RT 2 ) 


R = gas constant 
T p = effective temperature of particles, and at high temperatures T p > T, 

If now a particle, in this case an electron, travels a mean free path \, in the 
depletion region at a velocity v, the collision rate will be 


Collisions/unit time = — 


and the mean rate of energy gain per particle will be 

x) d£ 


This may be averaged for all particles so that the average rate of energy- 
exchange will be: 


Total gain 

all particles 

_Total number of particles 

J N(v)dv ■ de 





a i 2 2 
km u 

1 (2ltRT 2 } 



1 - 




2 3 

v dv 





2 2 

J Q (2«RT 2 ) 


e " - v^dv 

r*4 fsj 


2 2 

a ^ -Bv ,. -Bv 

(v 5 • e - Cv^e )dv 

= A 

>a ? -Bv 

v e dv 

A = 


B = 



C = 


Tt 1 2 

5 = o m ^ 

As shown in the Appendix (page 62) the integrals may be evaluated to give 


St- A • 

dt A 





l6kT u 

dt = 


Equation (6) shows that if 

T = T 
2 1 

no net transfer of energy will take place between electrons and the lattice, 
i.e., thermal equilibrium prevails. 

2.3-2.4 Mobility Due to Lattice Scattering 

Knowing the rate of energy transfer between thermal lattice energy 
and the carriers in the depletion region, the influence of the electric field 
on the mobility may now be found. This seems best done with the help of the 
concept of effective temperature, where this temperature is a measure of the 

In the general case of electrons crossing a depletion region under 
the action of an electric field E, the rate of energy gain may be written as : 


Force X Average velocity ~ qp-E 


gained from E 

In (7), p. is the effective mobility and the relation above is only approximately 
true, since in a high field region diffusion may also play a part in carrier 
transport. For the present, however, only field effects will be considered; 
especially if the current is relatively small, will this be a reasonable 
assumption. A characteristic of microplasma currents on the other hand is that 


of extreme current density,, and this may require a diffusion correction to 

the drift expression. In the interest of simplicity, this will not be considered 

in the present analysis. 

Contemplation of the physical model assumed for current flow in the 
depletion region and the interaction between the carriers and the crystal lattice 
makes it apparent that the direction of energy transfer, obviously determined 
by the relative effective temperatures, is not immediately obvious. 

The very simplest case will be that where the energy gained by the 
carriers from the field is all dissipated in collisions with the lattice, i.e,, 



+ 3t 
from E 

- (8) 

from lattice 

Equation (8) indicates an equilibrium condition- -a steady temperature will have 
been reached and no net transfer of energy takes place. This obviously is a 
first-order assumption, and does not take account of the impurities. However, 
a solution which does not make use of this kind of simplification becomes 
prohibitively complicated with a seemingly small gain in accuracy. 

Smith has shown that by using this simplification the effective 
lattice mobility may be expressed as a function of the low field mobility n_ 
and as a function of the ratio of the lattice to effective electron temperature, 
in the form: 


The relation between the two temperatures has been given as a function of the 

electric field E, by Shockley as : 


which may be solved for I — I in the usual way to give 



1 + 

3« {»<#¥ 


In the presence of very high fields, like those encountered in junctions where 
microplasmas form, the following inequality holds: 

S» T. 

From (9) and (10) the mobility due to lattice scattering in the depletion region 
is thus found to be 

-V 3n 

H-J 3 

i-^ u 


2.3«2.5 Mobility Due to Impurity Coulomb Fields 


Either the classical analysis of Mott and Massey or the more recent 

Ik , v 15 *6 

work of Conwell (and Weiskoppf ) or that of Dingle and Brooks may be applied, 

but their results are not widely different. 

The first two analyses mentioned lead to what is now commonly known 
as the Conwell-Weiskopff formula^ the remaining analyses have in common with 
the former that they lead to equally unwieldy expressions . 


However, using values given by Conwell: 

u. = 8.5 X 10 






1 + 8.3 x 10 1 

8 if 






The mass ratio — has been determined by cyclotron resonance methods by several 

workers, and for electrons in S., this value may be taken to be about 0.^, 


N. = 10 15 /cm 3 

T = 78 K (liquid nitrogen) 

and for equation (ll), the parameters may be assigned values 

u n = 0.135 m /volt sec 


u = 5 x 10 m/sec 

E = 10 7 v/ 


The two mobilities thus are computed to be 

\in = 0.01 m /volt sec 

u. = 0.4l m /volt sec 

and they may be combined by the approximate relation: 

± = ^ + ^ (13) 

whence : 

u = 0.098 m 2 /volt sec (1*0 

It is important to note from the above that according to this approach, the 
impurity scattering greatly reduces the effective mobility and furthermore 
that the lattice mobility decreases rather sharply at high values of depletion 
layer field strength. 

2.4 Transit Time 

Knowing the carrier mobility and the dimensions of the depletion 
region, the transit time of charges across the high field region may be computed, 

The transit time, T, is given by; 




v = uE 

From the values quoted above, follows: 

7 6 
v = 0.098 x 10 ' 2 10 m/sec 

while in the units presently under test, the depletion layer width, at breakdown 
is about 

W = 10 meter 


m 10_6 1 • 

T = — t~ - 1 picosecond 

10 6 

Transit times of this order of magnitude have been predicted by Salzberg and 

Sard, although no calculations were given. 

2.4.1 Voltage Influence on Transit Time 

It is interesting to note the probable variation of transit time with 
applied voltage and the possibility of decreasing the charge transit time by 
increasing the junction voltage „ 

It is assumed that the impurity scattering mobility may be considered 
independent of the field but the lattice mobility as mentioned decreases with E. 

In particular, 

H = *■ ' 


where K, is a constant, and from equation (ll) this gives: 


From equation (1*0: 

35.1 2/ ^ 
[in - ~ — m /volt : 

u = ■ ~ ■ r ) m / volt sec ^5) 

v& + ^i V35 + u.n/e, 

In equation (15) it may be noted that the fields presently under consideration 

are larger than 10 v/m so that with a p.. of 0.4l 

u.\Te » 35 
l ~ 

whence from equation (13): 

|i ~ • — m /volt set 


The velocity of transport thus follows as: 

v = |iE = 3^/e 


At this stage it should be recalled, though, that the field E is not only a 
function of the applied external voltage^, but also of the width of the depletion 
region, i.e., 

~ W 

where W = Kjs/V and L is a constant, 


K 2 



v = kVv 

The transit time, T, which is expressed by w/v, will thus take the fo 

rra : 

=r— = K^V (16) 

From equation (l6) it is seen that a high voltages no gain in transit time is 
accomplished by increasing the applied voltage. In fact, considering the non- 
linear relation between mobility and field strength, it may be possible to derive 
an optimum applied voltage for a specific junction in order to ensure minimum 
transit time. The law relating W to V for the specific junction will, however, 
first have to be determined for every specific case. 

Due to the approximate nature of this type of analysis a word of caution 
regarding interpretation is in order. An extension of the ideas leading up to 
equation (ll) illustrates this point. 

Smith, for instance, proves that by using the ideas mentioned in 
the preceding paragraphs, the high field mobility may be expressed in terms of 
the low field mobility according to the expression: 

[i = uJl - aE ] 


3fl .2 


a '- — 2 ^0 

Using the approximate values of 

u = 0.135 m /volt sec 
u = 5 X 10 m/sec 


it follows that 

H -0 


E -* 10 5 v/ 


As this is not supported by experimental observation it is obvious that the 
model is restricted in its validity. 



The results derived above apply to the behavior of an average carrier. 
It remains to be determined what spread there may be in time if a charge packet 
crosses the depletion region. If this spread is appreciable the current build- 
up in time will be adversely affected at the output of the device. 

In order to make analysis possible, it is necessary to simplify the 
model somewhat. 

It is assumed that a depletion region of width W is present. The 

microplasma discharge is now assumed to occur in the center of this region and 

to have dimensions negligible compared to the depletion width. 

This result is partly supported by results reported by Chynoweth who 

has found that in a depletion width of W, where the ionization coefficient is 

a.(E), the multiplication factor M follows the law: 


1 - h ~ 0.3a. (e) • w 

indicating that ionization takes place only in about one-third of the depletion 
width. Although being by no means negligible compared to the depletion width 
the "source" of charges does seem to be a localized effect. Furthermore, since 
the ionization coefficient increases with electric field strength maximum 
ionization is more likely to occur where the field is strongest, i.e., in the 
center of the depletion region. 

Considering also that both holes and electrons have to be accelerated 
before achieving ionizing energies, it is conceivable that ionization will not 
take place near the depletion layer edges where carriers have just entered the 
high field region. This implies also that carriers entering one end of the 
depletion region will not require the total depletion width for accleration to 
ionizing energies due to the factor "0.3" mentioned above, 

In this simplified model then, an infinitely narrow charge packet now 
starts at x = W/2, is acted on by the field, and it is required to know what its 
dimensions are at x =• W. 


This problem is well known in the study of electron packet propagation 
in vacuum tubes but it should be remembered that in the present case there are 
two significant differences between this and the electron packet problem. First, 
the microplasma essentially creates two charge packets travelling in opposite 
directions, while the transit of the field region now also is not in a vacuum 
but through a medium containing atoms capable of yielding free carriers when 

The spread in charge distribution in space is smaller for small 
mobilities than for larger ones, and the biggest spread will thus occur for the 
electron packet . The more diffused packet will probably take the longer time 
to traverse the edge of the depletion region thus being the slower rising current 
pulse; (this however is also affected by the ratio of the mobilities of the two 
types of carriers ) . 

3=1 Electron Current in Depletion Region 

In keeping with the idea of a simplified model a constant electric 
field will be assumed in the depletion region. The continuity equation for 
electrons, neglecting recombination in this high field region, will be: 


3? = D • f n (17) 

Noting that "n" here signifies the deviation from the equilibrium level of 
electron density, this equation is seen to be identical to the general heat 
flow expression. 

If at time t = 0, the flow starts, while the electron packet has 


negligible dimensions, the standard result from the theory of heat flow 

indicates a solution at any subsequent instant t, as: 


N K 2 t 

n = — — e 


where k and k are constants and N is the total number of electrons in the packet 


For the case of electrons moving in a semiconductor, the constant,:; k 
and k may be evaluated to have n read 

N e~^ (18) 


However, contemplation of equation (l8) shows that the influence of the electric 
field has been neglected, while it is intuitively felt that its influence will 
be to increase the x-coordinate of the (center of the) charge packet. 

If x is the true coordinate at time t, while it would have been x ! in 
the absence of the field, one may write: 

x = x' + uEt 

As for the solution for n in equation (l8) however, the packet will only have 
diffused the effective distance x', so that x in (l8) should be replaced by: 

x* = x - uEt 

whence : 

(x-uEt f 
-= e U9) 


Consider now the shape in x, of n at any instant to Let the pulse have moved 
from x = 0, where it started at t = 0, to some point where it is momentarily 
frozen so as to observe its shape. 

Equation (19) shows that the spread of the pulse between its (l/e) 
points from the central peak is of the order of 

2x \lk~Dt = Zsc 
Consider now the order of time taken to move this distance (Fig. 3)« 


Figure 3 

First assume that the pulse has moved from the center of the depletion 

region to its edge- -the previous analysis has shown that this takes a time of 

about 10 seconds to happen. 

Ax is thus found to be : 


Ax = k n/35 X 10" x 10" 12 ~ 2h x 10 meters 

The mean drift velocity had been found as 

v = 10 m/sec 

so that the time for the pulse to move across the edge of the depletion region 
from one (l/e) point to the other is: 


., Ax 2k x 10" ' . - nh . , 

At = — = 7 — < 0,24 picosecond 

~ 10 

This result indicates that although an average carrier will take about one 
picrosecond to cross the depletion region, about 25 per cent more time will be 
required for the total pulse to pass due to spread in particle position,, 



It is believed that thermal influences are of the utmost importance 
in junction breakdown, especially in microplasma-type breakdowns. This will be 
referred to again quite extensively at a later stage, but a brief indication 
of its probable importance will be in order now. 

Microplasmas occur in present-day Si junctions at voltages typically 
of the order of 20 to kO volts. The current carried by a single microplasma 
has been reported by several authors to be in the 10 to 200 microampere range. 

Take as a representative case 

V = 30 volts 
I = 50 microamp 

The rate of dissipation will be V X I = 1500 microwatts. 

A "typical' 1 "ON" time for a microplasma does not exist, but take as 
an example the practical figure of a discharge that remains on for one 

The total dissipation will thus be 

1500 x 10~ 12 joules = 1.5 X 10" 9 joules 

This is then the energy expended within the volume of a single microplasma y 

The actual physical dimensions of the active discharge region are 

1 o 

still somewhat in doubt but it is reasonable to assume from published data. 

that this region has a cylindrical shape of length equal to its diameter, equal 

-l ft 
to one micron. The microplasma volume thus is about 10 cubic meters, The 

specific gravity of Si is about 2.33 whence the weight of the microplasma 

volume is of the order of 

2.3 X 10" 15 kgm 

The specific heat is quoted as 760 joules/kgm C so that the dissipation of 

1.5 X 10 joules, could, if neglig: 

raise the temperature by AT, where : 

1.5 X 10 joules, could, if negligible heat were lost to the surroundings, 


AT ~ 900 C 

This temperatures which is of the order of 75 per cent of the melting point of 

Si seems very high, but could thus be approached, depending upon the thermal 

time constants of the regions involved . It should be remembered that the 

temperature rise above was the consequence of a single current pulse. Depending 

on the pulse repetition frequency, higher temperatures are conceivable „ Rose 

calculates an approximate rise in temperatures of about kO C but stresses that 

this may be conservative,, He assumes a thermal time constant of about 10 sees. 

Considering, however, some considerable experimental evidence on microplasmas 

which has become available, some aspects of the thermal behavior of microplasmas 

are far from clear. For instance, the original microplasma model of Rose assumec 

a high field in the regions flanking the actual discharge volume^ due to high 

ionization density in the latter volume, the voltage across it could not be very 


1 ft 
Shockley's model looks somewhat like Rose's and he calculates a 


space charge resistance of the order of 10 ohms but with a microplasma current 

of 100 microamperes, this accounts for only one volt of the applied voltage „ 
Shockley also calculated the spreading resistance of the two regions bordering 
on the discharge, but finds this bulk material resistance an order of magnitude 
less than the space charge resistance. 

It seems that the only probable solution could be that either the 
space charge resistance seems higher than predicted with the present model, or 
that rather severe deviations from Ohm's law occurs in the breakdown region. 
Keeping in mind these possible effects, it is nevertheless felt that,, in the 
light of experimental evidence (see paragraph 6) thermal effects play a much 
more important role than thus far ascribed to them. 



5.1 Observed Behavior 

Considering the current-voltage relation of a junction showing clear 
microplasma behavior, the most striking features, initially, is the very "hard" 
characteristic (Fig. k). 

Typically a reverse diode resistance of 1000 Megohm or more is found 
below breakdown; at and just above breakdown the resistance is difficult to 
determined, since the current pulse shape is still in doubt . Oscillograms 
indicate, however, that this latter resistance is relatively high, of the order 
of tens of thousands of ohms. 

5.1.1 Current-Voltage Relation 


One of the most complete discussions of micrplasma V-I characteristics 


has been given by Haitz based to a large extent on the model proposed by 

This model has among others, the following characteristics: if the 

junction voltages rises above a certain critical value V , the microplasma may 
switch on. On the average, it waits a time t before doing so and carries a 
current I. When the voltage drops below V , the microplasma switches off 
immediately . Two points may be raised here; the first is the abrupt switching 
off. Although both "on" and "off" switching has been found to be fast, the 
"off" time invariably is longer. That this effect may have a thermal origin is 
suggested by the experimentally observed decrease in switching off time at low 

ambient temperature (Fig. 11 ). However, if recombination plays an important 

role in switch-off as suggested by Jonscher, this process could be fast. 

The second point, and one of considerable practical important is the 


fact that Champlin, and others, have observed an average time lag t_ after the 
critical voltage has been reached, before the breakdown sets in. 

In any fast application this delay, which has been claimed to be of 
the order of a microsecond by Champlin, would render the phenomenon quite 
useless. It is thus essential that this delay must either be eliminated, or 
for present applications, reduced to at most say 1 picosecond. It is thought 


Figure k. Si p-n Junction V-I Characteristic 

Horizontal: k volts/cm 
Vertical: 5 microamp/cm 

(Tektronix Transistor Curve Tracer 
at 2^0 c/sec) 



that this delay is caused by the fact that although a sufficiently high Pie] 
may be present in the depletion region, the arrival of an initiating carrier 
has to be awaited. 

It is contended now that the discharge will appear immediately, even 
in the absence of an externally derived initiating carrier, if a carrier could 
be released within the confines of the depletion region. This could be done 
by sufficiently raising the applied voltage to release carriers by field 
emission, or by injecting photons to energize potential ionizing carriers. 

The likelihood of this being the case is supported by the observed 


variation of the function ft (Champlin ) from to infinity over some small 
voltage range; this means that at some value of the field the discharge cannot 
extenguish since initiating carriers are always present. That it might be 
wanting to extinguish may be illustrated by the high noise content of this 
current. The heat generated by the current pulse cannot be neglected „ Rose 
calculated the thermal time constant of a microplasma region to be of the order 
of 10 seconds,, so that no sluggishness need be introduced by the thermal 
action, on the present time scale. 

Of particular importance is the fact that in a bistable microplasma 
the turn-off probability sharply decreases with breakdown area, i.e., with 
current. Since the AV's involved seem small, a change of AA in area would mean 
a change of (constant AA) current but a (constant X AA) in dissipation^ it 
might thus be that the higher local temperature keeps the discharge going by 
supplying carriers. This view is further supported by the slope of the turn-off 

probability vs. current curves. In Fig. 5 are shown some approximate curves of 

19 3 

results obtained by Haitz and by Mclntyre. 

Considering the three units shown, it is obvious that at higher 
currents, when the ambient temperature will be higher, the turn-off probability 
is less dependent upon current. 

Another point, however, is that higher ambient temperature, at fixed 
depletion layer field (or rather, fixed applied voltage), tends to extinguish 
the microplasmas . This is a very marked effect as the following figures will 
illustrate: A silicon junction breaks down completely (i.e., $) n is zero) at 
25.1 volts when the ambient temperature is 300 K. When this same junction is 
at 360 K the same condition is only obtained at 25.9 volts. This conforms with 



10 ' 



















10 3 






10 J 

10° I. 








25 50 


100 125 150 175 200 225 250 


Figure 5. Approximate Turn-Off Probability as Function of Junction 
Current (Haitz, Mclntyre) 


Mclntyre's idea that ionization probability decrease:; with increased lattice 
heating. It seems as if the higher lattice heating increases the lattice 

scattering cross-section, thus reducing the mean free path and hence the energy 

gain by the carriers per free path. Tokuyama, however, found that breakdown 

voltages could either increase or decrease with increasing temperature, 

Mclntyre mentioned in his analysis the possibility of extinguishing 
of the discharge by the decrease in mean free path, but it is evident at closer 
observation that several further factors may play an important role. 

After a discharge has been initiated, both space charge and spreading 

resistance, as well as compensating recombination tend to stabilize the current. 

Jonscher writes for the ionization rate 

a.n[N D - (N A + n)] 


a. is the ionizing coefficient 

N_ - (N. + n) is the density of ionizable impurities 
n is the total density of free carrier 

The recombination rate is considered dependent upon three recombination processes 
viz,, recombination accompanied by phonon emission, by photon emission and by 
Auger (impact) recombination. The first two processes are considered strongly 
temperature-dependent^ increasing sharply as the temperature drops, Among other 
things this means that although the mean free path decreases with temperature, 
the number of heat -generating collisions (phonon recombinations) also increases 
with lowering temperature. It will be evident that with the limited data 
obtainable from external observations on the integrated effects of all these 
processes, it will not be profitable to further speculate at this stage. 

5 *2 Electrical Equivalent Circuit 

An electrical equivalent circuit for the process under investigation 
is a necessity for successful application. The original circuit as proposed 


by Rose is shown in Fig. 6. A description of the breakdown mechanism and 
subsequent "charge state" of the depletion region leads to a model closely- 
analogous to gaseous breakdown (Fig, 7)» A severe shortcoming here, however, is 

the inherent negative resistance characteristic implied^ while there is increasing 

doubt about the presence of any negative resistance at alio 


A later model by Champlin, where the initiating voltage V is 


followed by a voltage interval dV where the breakdown seems unstable and above 
(V + dV) becomes stable again, has had some success. As pointed out by Haitz, 
however, this model describes quite well only the behavior in high impedance 
circuits but fails when the microplasma junction is in a low impedance circuit „ 


One of the most recent models is that by Haitz with this model 

containing the largest number of independent parameters viz., extrapolated 
breakdown voltage V (Fig. 8) series resistance R , turn-off probability.. 


It is significant though that although mention has been made of the 

possible importance of temperature on the process to be described, it has not 
explicitly been incorporated in the model „ Also, the insistance of many authors 
on the essential "chance variation" in junction current to explain extinction of 
the discharge, does not always seem plausable. It is thought that thermal 
effects play a much more important role than hitherto attributed to them and 
that in fact, the "on" and "off" characteristic can adequately be described by 
thermal considerations together with some randomness caused by the trapping and 
release of carriers in the depletion region . Some observed characteristics 
are however not explained by the models just mentioned „ 

First, just "at" the critical applied voltage, the observed breakdown 
current pulses are not all amplitude limited as predicted but vary widely in 
amplitude from practically zero to some well-defined maximum--see the top trace 
in Figo 9» (This trace, obtained with the same circuit as that on page 3,, 
represents a l/50 sec exposure of the film. ) It seems obvious that in this 
trace the smaller pulses have a shorter duration than the larger ones. 

Depending now upon the role played by temperature, two explanations are 
possible : 

If higher lattice temperature aids the breakdown process it could be 
argued that a small pulse generates too little heat to' keep itself going with the 
barely sufficient field strength* The larger pulses then obviously are of longer 


Equivalent series 

Field induced 

Figure 6. Electrical Equivalent Circuit (Rose) 

Hole current density 

Electron current 

Field configuration 
after breakdown 

Initial field 

Figure 7. Electrical Model (Rose) 




Supposed V-I characteristic 

Lmaediately Junction switches 
onto current 3L 

Due to (I x K, ) drop, V and 
thus I drop (with RC tins) 
to © 

V applied and Junction 
breaks down when initiating 
_, carrier arrives 


*—• Current sits at Q) untilchance variation of 
current turns it off to © (i.e., current falls 
to aero, whereafter, discharge disappears). 
After (A), Junction voltage charges up tp V. 
again with R-C time constant R-C. At ® , 
average waiting time for next Initiating carrier 
is 1 uaec. 

Figure 8. Nicroplasoa Equivalent Circuit (Haiti) 


Time base : 
Top trace : 
Bottom trace : 

10 microsec/cm 
50 microamp/cm 
100 microamp/cm 

Figure 9 


duration due to greater FR losses . This explanation,, however,, does not give 
a reason for the "why" of small and large pulses , 

A different role for the temperature could lead to the following 
reasoning: this seems the more plausable explanation since it is readily- 
established that at fixed externally applied voltage,, an increase in temperature 
of a fraction of a degree Kelvin greatly reduces the number of breakdowns per 
second in a junction „ 

The argument then is : At a certain instant^ at a particular point 
in the junction,, the field is just barely sufficient to start a breakdown and a 
small current pulse ensues. This pulse is, however^ just on the verge of the 
possible due to the low fields and the least decrease in energy gained from 
the field- -this decrease being the result of shorter mean free path due to 
larger lattice vibration caused by ohmic losses—causes it to estinguish,, very 
soon after starting „ The larger pulses however represent the currents started 
by field conditions capable of initiating a heavier pulse and thus require more 
heat to turn them off. 

This point of view is further supported by the variation of pulse 
length with the applied voltage prior to conditions where the field is sufficient 
to keep the junction continuously on. The lower trace of Fig, 10 shows that an 
applied voltage about 50 mv higher than that applicable to the upper trace^, the 
pulses are considerably longer and also have much longer periods between pulses. 
In the light of the foregoing discussion the higher field then initiates a heavy 
pulse which remains on quite long before enough heat is generated to extinguish 
it. It is of course true that the larger currents generate much more heat than 
the smaller currents,, considering the squared current relation to the losses^ 
however,, as was experimentally observed by several authors,, the larger currents 
also have paths of larger cross-section so that the current flowing and the 
temperature rise will not be simply related. 

After the heavy pulse has virtually extinguished itself due to the 
heat it generated^ the lattice also takes longer to recover before allowing the 
next pulse as shown by the longer "dead" times. 

It is considered that the possibility of a thermal origin for the 
switching off of the currents is a much more likely phenomenon than the 
arbitrary "chance" variation proposed by Mclntyre, Particularly since this 
chance variation seems to require a momentarily decrease to zero of a current 
of up to 200 microamp; does its likelihood seem remote. 

Short ON-time just at breakdown, 

Time base : 100 microsec/cm 
Vertical: 100 microamp/cm 

Short OFF-time about 50 mv above 
initial breakdown. 

Time base 
Vertical : 

100 microsec/cm 
100 microamp/cm 

Figure 10 


The one unexplained phenomenon of the mechanism proposed above is 
the reason for varying field conditions giving rise to a tendency for large or 
small pulses to appear „ 

Figure 10 gives further evidence of the well known variation of 
ON- time with applied voltage. 

In essence the proposed model would then be the following: At a 
certain lattice temperature T, , a minimum electric field is required in the 
depletion region to impart sufficient energy for ionization to a carrier entering 
this high field region with zero velocity,, Except possibly at very high temper- 
atures^ all carriers entering the depletion region may be considered to have zero 
velocity compared to their subsequent velocity. 

Taking into account the mean free path applicable,, such a carrier, 
entering the depletion region is accelerated by the field and gains energy with 
time, the rate depending upon the collision rate with the lattice and the corres- 
ponding loss of energy to the lattice. This energy loss heats up the lattice,, 
increases the number of collisions by reducing the mean free path and thus also 
the net energy transfer from field to carrier is reduced; the discharge thus 
extinguishes. Due to heat conduction the lattice cools down practically 
immedicately, thus creating conditions where the next discharge may start <> It 
is thus seen that in spite of the presence of a sufficiently strong field as 
well as initiating carriers, the lattice temperature has to be below a certain 
value for the discharge to start- As soon as it starts it extinguishes itself 
by increasing the lattice temperature- -a type of squelching oscillation may thus 
be expected, the period being related to the thermal time constant of the 
junction region. 

5.3 Energy Considerations 

Before taking a closer look at some detailed experimental observations, 
it will be in order to make an approximate evaluation of the energy relations 
mentioned in the paragraphs above. 

On page 11, the rate of energy gain from the lattice was given by 
equation (7) as : 

dl = ^ 


Now the field in the depletion region will to some extent depend upon the current 
flowing since resistance will be present in both the diode itself and the supply 
circuit. It is thus conceivable that the field will drop as the current switches 
on, and even though the thermal effects may not extinguish the discharge a too 
low field may also be the cause; this will have to be included in the picture 

If a critical initiating field is assumed as was mentioned above, 
this field being E , the field at any subsequent instant t will be: 

~c at 

(-T— will be negative.) The mean rate of energy gain in the interval t will thus 


51 = ^ 

OE 12 


The rate of energy loss to the lattice had been previously found to be 

de i 

.OKI' U 

1 m ' 

dt ~ 


j 2k V 


As noted before, the effective carrier temperature T may be expressed as a 
function of the field in the form: 

T l 
T = -= 

2 2 



= -f [1 + K^E] 



3n ro ; 

VrW 1 


Equation (21 ) thus becomes 



V~2~ (1 + K ^ } [1 " 2 (1 + K i^ )] 












The rates of energy gain, given by (21 ) and (22), both multiplied by the time t, 
will give the total energy gained in the depletion layer. For determining the 
limits of microplasma formation this may be equated to the required ionization 
energy . 

The complete expression thus becomes: 



E + 

dg i2 
5t 2 

t + K r 

~ (1 + K h E) [1 - | (1 + K^E)]t > 6 



K h=- 

qw 2 


K 5 = 

l6ku £ 



In equation (23) ~ still has to be found, 

5.3d Determination of g— ■ 

Experimental evidence of microplasma behavior (e.g., Fig. l) shows 
that the current is amplitude limited „ This means that, no matter with what 
time constant the current rises, it tends to some final value in a very short 
time. It will be assumed that the rise in current is determined by some time 
constant T, and that the current as a function of time may be expressed by the 


1= I (l - e" t/T ) 

where I is the final value. In the depletion region the field E will be given 


E = 

V - IR 

where V is the applied voltage, R the effective series resistance while as 
before, W is the effective width of the space charge region. 

If it is assumed that we have an abrupt junction, as was the case in 
the units under test, the voltage and space charge width are related by: 




*~=h^-^ = h 

■ I Q (1 - e" t/T )R- 
siV - ■ — - 

The rate at which E changes with time will thus be given by; 



lV -t/T 




Equation (2k) may be used in (23) but the time constant T still have to be 
determined o 

The microplasma itself has been found by several workers to be about 
one micron in length, and the total charge in its length will thus be given by: 

Q = 

I X W 

and by using previously mentioned values of 


I = 100 ua 


W = 10 m 

v = 0.128 X 10 7 m/sec 

Q, = 10 coulomb 

If the voltage across this region is 20 volts, the capacitive effect associated 
with it is : 

C = & = 10" 1 farad 


With a possible series resistance of 10 ohms, the R-C time constant becomes 

T ~ 10 sees 

This result leaves only t as the unknown in equation (23) 
The term (l + K. ) in (23) may be written as: 

(k 6 + y^) 


Filling in the expression for -r— , (23) becomes: 


«e-"8 e 

-t/ T 

r^ ) 7777 


t + (Kq - K 1Q e 0/ ' ) s/K i:L + K 12 e ' • t > e_.„ 

— mm 

with the constants being given by: 



K i ■ w 

\-^(tT ^0" -1*5*7 

l6ku / m ' 1 

K 5 " ~X~ V2k^ *!! " 2 K 6 

K iW ?i 

v _ v — ± v 

^ " lWv" K 12 - - "7 

K W 

8 2t/v 

From equation (25) it will now be possible to find 

t = fd ) 

assuming all the other parameters to be constants . It might also be more 
appropriate to write up a computer program for finding t as a function of several 
of the parameters likely to vary,, like temperature. This is quite feasible at 
the present stage, but it is felt that quite considerably more experimental data 
of the microplasma type behavior should be gathered so as to act as guide lines 
in making approximations and thus evolving more practical equations. 

At this stage some remarks about the energy required for ionization 
and the process itself are in order. 

The true ionization energy required is not a readily determinable 

quantity. As mentioned by Jonscher the mean value theory has to some extent 

explained experimental results. This theory requires that the critical break- 
down field is only reached when the average energy of all carriers present 
reaches a certain fraction, y } of the "true" ionization energy--7 has been found 
to be of the order of 0.2. 

It seems though that the above reasoning points to a weak link in the 
efforts to reconcile theory and experiment in the case of breakdown phenomena. 


It is suggested that the value 0.2 is just a result of the fact that only at 
this value of 7 are there sufficient oarriers at ionizing energies to make their 
influence ohservahle . Limitations of oscilloscopes and related equipment has 
very often been found to he the determining factor in deciding upon threshold 
values for breakdown voltages. Extremely accurate experimental observations 
^ considerable period of time on several samples support the view that 
often some extent of breakdown already is present before the positively identified 
discharge is noted. This point was also commented on by Mclntyre and is 
further illustrated by the fact that as ever better junctions are produced, the 
equivalent circuit seems to be getting ever simpler. 

In order to obtain some idea of how the energy conditions of the • 
carriers in the depletion region would change with time, the equation giving • 
the energy gained and lost may be further examined. 

5 A Energy Gained 

,.„ f „r\ .,_., the energy gained by the carriers 
The first term of equation (25) gives tne enwBJ b 

from the electric field E, in the depletion region. 

The rate of energy gain with time was originally given as 


Ef 2 

so the total energy gained in time will be a function of both the field and of 

the time, say At, itself. 

Considering for the moment interpretation of the time, At: If at 

a v -> w where E is the minim-urn field to cause 
instant t = t Q one has T = T and E > E Q where ^ 

4. w m the time At will indicate how long the breakdown 
ionization at temperature T Q , the time zax w±xx 

may remain on. It is explicitly stated that it mav remain on for At, since 
^ re is no guaranty that the breakdown will start as soon as the voltage is 
applied, at the end of At though, the breakdown will cease. Trapping of course 
may also play a part in the behavior as has already been mentioned. 
Consider now the terms representing energy gained: 

qp[E n - E(t)] 2 • At 



The component, E(l ) of the field which varies with time is a result 
of the current, varying with time. (A second order effect, due to variation 
W will be neglected. ) 

Since the current starts at zero, this field component should start 
at zero and after infinite time will be equal to 

l o ■ R 



E(t) = 

V 1 

e" t/T )B 


and the energy gained in time At may be expressed by 





e ' ) 


if At is small. Using some of the values given in the Appendix, and considering 
At to vary between O.lT and 5t, Fig. 11 shows how the energy gained will vary 
with time. It is important also to note how the rate of energy gain changes 
with time, as given by the changing slope of this curve, the rate being seen 
to increase. 













Figure 11. Carrier Energy Gain from Electric Field 


5-5 Energy Lost to Lattice 

The rate of energy gain from the lattice was given as 

de = l6ku 2 / m ' \ 
dt \ 2 V2kT n T, 

and the carrier temperature T , expressed in terms of the lattice temperature 

T l 

T = — 

2 2 


Substituting for T and grouping terms one may write 

de K 5 



2 ^2 

^(1 +lc|T (1 - K^E) 

K 4 -7t hr 

v l6ku^ 
K 5 = ~AT 


In this expression both E and T, will be functions of time so that also the 
rate of energy loss is going to vary with time., 

5. • 5 • 1 Variation of T, with Time 

Rose has analyzed the heat distribution inside a probable model of 
a microplasma region, and by taking account of heat los to the surroundings, 
he finds that at a dissipation rate P joules/sec, the temperature increase will 



k is the heat conductivity in j/sec cm C 


I is the diameter of a sphere in which the 
dissipation is assumed to take place 


P- I 2 • R- I^CL-e- 1 ^) 2 

the temperature of the microplasma region, which is given by: 

T = T + AT 

will vary with time, increasing to a final value determined by IT • R„ 

As shown by the expression for AT, this temperature rise will not be 
a simple function of the dissipation, and as an approximation it will simply be 
written as : 

m = ^(1 - e" t/T ) 

where IC, is some constant. The lattice temperature T, should thus be replaced 


T x (t) = T Q + ^(1 - e" t/T ) 2 


5o6 Complete Expression for -v— 

Incorporating the expression for T.(t) and E(t) in the expression 
for the energy gained by the carriers, yields: 


^v^-' t/T ) 2 H 


i -t/T 
1 - e ' 



xii - K, 

E .V -t/T' 

~o w v ; 

This rate of energy exchange may be evaluated at a few points, such as shown 
below : 

At t = 0: 



Z = -4- Jt7 Vl + K,E n ' (1 - K.EJ 

sT 2 





At t - T : ^ z — v x Q - 

+ 0.1^1" n/1 - 

E - 

0.4l Rl 

~o w 

x - w^-f) 

when t ->• K : t- 



Vt + k„i? V: 

3t-^ T + V0 

* w 

1 - K, 


Inspection of the above expression shows that the rate of energy exchange decrease 
with time, and furthermore is of such a sign as to indicate energy transfer from 
carrier to lattice. This agrees with the fact that at the fields under considera- 
tion the effective temperature of the carriers will be much higher than that of 
the lattice 

Sketching out the variations in (rrr.) with time as arrived at above, 
for both the energy gained from the field and that lost to the lattice, some 
interesting points come to light. 

Consider first Fig. 12 where on the same axes is shown the variation 
in energy gained and energy lost by the carriers-- the rate of energy gained is 
seen to increase with time, while the rate of energy lost decreases with time. 


It is tacitly assumed in this first case that the energy gained is more than 
the energy lost at t = 0. 

The net gain in energy by the carriers will be given by the difference 
between the ordinates of the curves--this difference is plotted in Fig. 12. 





t = 



Figure 12, Carrier Energy Gain and Loss 

Depending now on the value of the minimum required ionization energy,, 
different conditions in the depletion region may prevail. Consider Fig, 13 
where the dotted lines indicate different values of the difference between 
energy gained and lost—this latter difference will have an absolute value 
determined by applied voltage, initial temperature, etc. 

The value, £., indicated below is the minimum required ionization 
energy, its value depending among other things upon the material being used. 

Curve A indicates the case where the applied field would be too weak 
to cause any ionization at all during the time the carrier spends in the high 
field region. 

Consider then Curve B: If the applied voltage is so high as to give 

e, as the initial net energy gain, ionization will result, and the microplasma 

will switch on. It will then remain on until time t n when the net gain in 


_2, 7 - 













Curve C 





Curve B 

Curve A 





Depletion Layer Transit Time 

Figure 13 . Varying Conditions in the Depletion Region 

energy will fall below the required ionization value and the discharge should 
extinguish . However, at time t the energy gained will exceed the energy lost 
by a sufficient amount to re- ignite the discharge. 

Curve C will represent the case where the applied voltage is so large 
that at no time during transit of the depletion region does the energy fall low 
enough to extinguish the discharge. 

Apart from any trapping effects it will be obvious that a certain 
randomness will be introduced by the energy with which a carrier enters the 
depletion region. 

Apart from statistical cons iderat ions 3 a very detailed observation 
of delay times and pulse lengths as functions of applied voltage will have to 
be made to provide experimental proof for the arguments above. 

In Fig. Ik below are shown some current pulse distributions in time, 
as would result from differing initial energies for carriers entering the high 
field region. 





Figure ik. Current Pulse Distributions in Time 

It is conceivable that, instead of the initial conditions being 
greater gain than loss in energy, it could be the other way round „ This means 
that a carrier entering the depletion region would for the first short time 
interval lose more energy to the lattice than it was gaining from the electric 
field. Due however to the increase in rate of energy transfer from the field 
accompanied by the decrease in the rate of loss to the lattice, the carrier can 
still exceed the critical energy while in transit across the high field region.. 

Da this case, the curves of Fig» 15 would apply, they being analogous 
to those of Fig. 13, 

-k 9 - 







Energy Gain 

e . 
_ _ J: +. 


Net gain 







Figure 15 » Alternative Condition for Carrier Energy Gain and Loss 



6.1 General Behavior 

Equation (25) illustrates quite adequately that an accurate theoretical 
description of microplasma breakdown does not seem quite possible as yet. Many 
constants required for quantitative analysis are completely unknown, so that at 
best, a semi-empirical solution will be available. 

One of the unsolved problems has always been the true shape in time 
of the microplasma current pulse--the time order discussed in paragraph 2. it- 
illustrates why experimental measurements are very difficult indeed. 

In the sections following some breakdown characteristics of 
microplasma-type silicon diodes are shown. These diodes, of especially accurate 
construction, were produced for noise production by the Shockley Laboratories 
of the Clevite Company and generously donated for the present investigation. 

6.2 Experimental Set -Up 

Several workers have found that junction breakdown may be precipitated 
in a junction (where a near-critical field is present) by the injection of 
photons from outside. It was thus decided to stress a junction electrically 
with a steady dc field to near its breakdown value and then to inject a periodic 
light pulse to start the breakdown and give a recurrent microplasma coincidental 
with the light pulse. 

The sketch in Fig. l6 on the next page gives an outlined of the 
experimental set-up. 

The light source is a P.E.K. 109 high-pressure dc mercury arc lamp 
producing a 0.012-inch cube arc, of 100 watts dissipation. The revolving disc 
of opaque cardboard was attached to a 6- inch diameter brass flywheel, driven 
by a l/l6 hp ac-dc Bodine motor at speeds up to 6000 rpm. 

The light from the source is first focused onto a fine slit in the 
disc; thereafter it is again focused onto the junction at J. Some light spills 
past the junction and is picked up by the photomultiplier PM which delivers, 
via a three-stage amplifier, a trigger pulse for the pulse generator PG. This 
pulse generator was then employed to give a trigger pulse for the oscilloscope. 



















i § 


O 0> 
M S. 

8 t 

o a 


i - 


















§ * J 

£ 0> 

a 3 


-d >d +» 
<*"»cvj evi 




The actual voltage pulse from the pulses of microplasma current is 
;en across the resistor R in series with the diode. 

6 „ 3 F iiv.t Experimental Results 

In Fig. 17 on the next page are shown some oscillographs of microplasma 
breakdowns . 

Photo (a), which was taken at room temperature, shows that although 
microplasma pulses are present all the time once the potential has reached a 
certain value, many more breakdowns are produced when the ligh pulse reaches 
the junction; this would indicate that photons aid in the production of ionized 
regions . 

It is significant though that although a steady stream of photons 
reaches the junction while the light pulse is on, the breakdown is not continuous-- 
this is clearly seen in the close-up of photos (b ) and (c). 

It is possible however to eliminate spurious breakdown between light 
pulses by shielding the junction from ambient light and cooling to liquid 
nitrogen temperature „ Fig. 19 on page 55 Shows clearly that between light 
pulses no other breakdowns are present or that they are at least so weak or fast 
as to be not observable on the oscilloscope in spite of the long exposure (3 sees) 
used for photo , 

All the evidence cited above indicates that thermal influences play 
a major role in the formation of microplasma breakdowns, as had been mentioned 
during the foregoing analysis. 

A further significant point is that as is indicated in Figs. 18a and 
18b the ruling temperature certainly influences the rise and fall times of the 
current pulses. Figure 18 shows that although the rise time is sharp, at room, 
temperature, the pulse dissipation increases the ambient temperature and when 
due to this rise in lattice temperature the breakdown extinguishes, there follows 
a relatively long fall time. At liquid nitrogen temperature Fig. 18 indicates 
that when the lattice is continually cooled, the fall time approaches that of 
the rise time. 

In an attempt to find the true pulse rise time, it will be obvious 
that the conventional oscilloscopes employed above are much too slow, considering 
the expected rise times of picoseconds. 


base: 2 msec/cm 
Vertical: 100 |ia/cm 
(expot ecs) 

(b) Time ba 



(c ) Time base 

5 |isec/cm 
100 ii a/ cm 

Figure 17 

(a) Time base: 2 |asec/cm 

Vertical: 500 (la/cin ac I Kfi 
(Oscilloscope passband 100 mc/cm) 

Room temper at ire 

(b) Time base: 200 nsec/cm 

Vertical: 100 fia/cm across 100 ft 
(Oscilloscope passband 10 mc/sec ) 

Liquid nitrogen temper 

re 18. tee of Temperature on Rise and Fall Times 


Horizontal: 5 msec/ cm 
Vertic- 50 pia/cm 

(exr fie 3 sec) 

Fig ire 19 


The only truly fast conventional oscilloscopes are those like the 
Tektronix 519 with a 1000 mc/sec passband- -however, since, this type of 'scope 
employs a travelling wave type CRT and in order to obtain the mentioned response 
cannot incorporate any amplifiers, the low sensitivity is prohibitive „ Con- 
sidering that the 519 had a sensitivity of 10 volt/cm and a microplasma pulse 
has a typical current of 100 microamp, a 10 kilo-ohm resistance is required in 
series with the diode to provide a vertical deflection of 1 millimeter on the 
screen. This high value of resistance, together with the stray and input 
capacitance of the oscilloscope immediately limit the current pulse rise time to 
something of the order of a nanosecond or more, thus defeating the purpose of 
the fast oscilloscope. On the other hand, using the 125-ohm input impedance of 
the 519 & s a diode load produces a deflection too small to be detectable- - 
considerable effort was directed at this latter method, hoping to photograph a 
small deflection and then use photographic enlarging for greater detail. Although 
the 519 has a very fine trace, no success was achieved with this method. 

The only other alternative at the present time is the use of a sampling 
oscilloscope. In order to do this, the signal to be sampled must, with present- 
day sampling oscilloscope like the Tektronix 66l or the 3S?6 and 3T7T plug-in 
units for the 56IA be of a strictly periodic nature. In the case of microplasmas, 
it is obvious from Fig. 17a that this will not be an easy requirement to comply 

The photon bombardment, showing the increased breakdowns in Fig. 17c 
suggests however that this method might be used to provide a periodic pulse. 

6.4 Attempts at Using Sampling Techniques 

The circuit used was essentially of Fig. l6 but the oscilloscopes 
used were, first, a Tektronix 66l with a 4S1 sampling head and a 5T1 timing 
unit. The current measuring resistance R was eliminated since the 50-ohm input 
impedance of the sampling unit served this purpose. 

Variable delays were available in both the trigger and signal circuits 
but despite extensive efforts it was not possible to obtain a reliable trace 
of a single breakdown pulse. Figures 20a and 20b show photographs of sampling 
'scope traces. These traces show up some interesting phenomena, even though 
they do not reflect the true current pulse shape. 




Figure 20. Sampling Oscilloscope Traces 


The follow lag should be stressed at the outs< displa 
traces of Fig. 20 the sampling oscilloscope was being triggered by the chopped 
light signal via the photomultiplier and amplifier. The rotating disc caused 
this triggering signal to have about a 10-millisecond pulse spacing, so that 
each dot on the trace is separated in real time by this period. Furthermore, 
the three traces were taken at about one-minute intervals. It is thus remark- 
able that there is such a correspondence between the traces indicating a def in I 
pattern in the breakdown current waveshape. This may suggest that, despite the 
present failure, periodic pulses may be produced in these junctions. 

The traces show a definite sequence of three breakdowns after which 
it seems that there is a slow decrease in pulse amplitude before the pulses 
decrease altogether, to restart the cycle. 

6.5 Delay in Switch-On after Simulation 

It was mentioned in paragraph 5 that Champlin had found in his 
work that an average waiting time of one microsecond elapsed after application 
of the breakdown voltage, before the first microplasma appeared. 

When considered in detail, this is not an easily determinable quantity, 
for several reasons. First, if the applied voltage is slowly increased either 
smoothly or in steps to the point where breakdowns are evident, it is practically 
impossible to say at what point the first pulse appeared since a definite change 
in junction characteristics with time is present. If the junction is electrically 
stressed for several minutes to just below "breakdown, " and the voltage is then 
increased, this effect is somewhat smaller, but it has not been possible to 
give a quantitative result in this respect. 

If pulsed voltages are used, as was discussed in DCL Report No. 1^9 
these same long time constant effects as well as capacitive surge currents 
obscure the results. Also, if a nearly large enough dc bias is applied, and a 
small pulse is superimposed, the condition is improved but suffer from the same 
disadvantages. At these low levels it must be remembered that where these 
fast pulses are concerned, matching problems become severe. The microplasma 
current is only of the order of 100 microamp and in order to prevent circuit RC 
times from swamping the phenomenon under test, the series resistance for current 
measurement must be low--say 100 ohms. This means a signal voltage of 10 mv 
only is available which means that observation is confined to slow oscilloscopes. 


Last, but by no means least, come the limitations of the junctions 
under test. These junctions, as often mentioned, change their response slowly 
(on the present microsecond time scale) and react differently to dc potentials 
and to pulses. The present junctions, constructed extremely carefully and from 
the purest material available, show much less change than the transistors of 
Report No. 1^9 but still render impossible the time delay measurement. 

Figure 21 below shows breakdowns occurring about ^-00 to 800 nanosec 
after application of the pulse--for the present junctions this is a representative 
figure although both shorter and longer times were observed. 

Horizontal: ^00 nsec/cm 

Vertical: (bottom trace) 100 jia/cm 

Figure 21 



The very fast current pulses produced by high electric field breakdown 
in Si p-n junctions have been recorded oscillographically. Pulses of hundreds 
of nanoseconds are easily observed, and are in the 100 to 200 microamp range. 
Faster pulses of comparable magnitude are visible but is has not been possible 
to record an individual fast (nanosec or less) pulse, probably due to the still 
random occurrence. 

All lengths of pulses seem available, from spikes too short to be 
reliably observed on a 5 nanosec rise time oscilloscope (Tektronix 585) to 
pulses of millisec length- -pulse length is smoothly variable by variation of 
applied voltage. The voltage range effecting this large variation in pulse 
length varies between diode samples but is of the order of tens of millivolts. 

Highly increased pulse repetition rate occurs under photon bombardment 
from a mercury arc source . 

It has not been possible to determine the delay between the application 
of the light pulse and the appearance of the first discharge—this was due in 
part to the relatively slow rise time (5 microsec) of the light pulse and the 
fact that it is not known how much external light is needed to precipitate a 
breakdown. (Commercially available fast light pulses produce about 2 x 10 
photons per 2 nanosec pulse, which after practical experience with the mercury 
source, is considered too weak. This undoubtedly is due partly to the fact 
that these diodes under test were encapsulated in glass, making focusing on the 
diode wafer very difficult. ) 

Consideration of the breakdown mechanism, as well as detailed 
experimental observation, suggests that the dominant parameters for breakdown 
are twofold: 

Electric field strength 
Lattice temperature 

It is believed that all other parameters can be expressed in terms of these 
two and it is felt that the prime target of microplasma research should be 
directed at reducing the number of parameters used to describe junction behavior 
to the above two. 


It is believed that positive control over breakdown pulses will 
become possible in the near future when even better p-n junctions become 

The easy control over average pulse duration holds promise in fast 
logic circuits as suggested by Professor W. J. Poppelbaum. 

Suggestions for Next Steps in Research 



Express turn-on and turn-off probabilities as functions of 
lattice and of carrier temperature. 

Obtain junctions not encapsulated and with ample optical access 
to both sides of the junction. These should preferably be Si 
structures since Ge is much more opaque to readily available 
light sources. 

Construct dc source of utmost stability and resetability with 
"zero" internal impedance. Available voltage should be about 
kO volts. 

Repeat experiments described in this report but with better 
controlled environment as regards light and heat. 

■ 62- 


Evaluation of Integrals 

Equation (5) read: 

°\ 3 -Bv 2 ,_ 5 -By£ Nj 
(v J e - Cv y e )dv 

ie - A ° 

dt f" 2 -Bv 2 

/ v • e ~ dv 

Consider the integral 

3 -Bv 2 
v e dv 


v = x 

2v • dv = dx 

. . dv = — 

~ 2v 


Integral =»—""./ x - e dx 

The solution to this integral is 

i r(n + 1) = _1_ 

2 B n+1 _ 2B 2 

with n = 1. (C.R.C. Standard Math Tables, C.R.C. Publishing Co., 1962o) 


By a similar change of variable the second integral may be evaluated 


i C / 2 -Bx, C n! C . 

Integral = — / x • e dx = — — — - = — when n = 2 

2 J 2 B n+1 B 3 


J v 2 e _Bv dv = ^ -s/n/B 



1 Wolfendale, A, Brit. Proc. IEE , 1957- 

2. Salzberg and Sard. Proc . IRE (letter), Oct. 1957. 

3. Mclntyre, R. J. Jo of Appl. Phys ., June 1961. 
Uo Henebry, M. Journ„ of Sc, Instruments , i960. 
% Haitz, et al. J., of Appl, Phys ., June 1963. 

6. Rose, D. J. Phys. Rev. 105 , 1957 

7. Ruge and Keil. J. of Appl. Phys ., Nov. 1963. 
80 Champlin, K. J. of Appl. Phys . , July 1959= 

9, Chynoweth, A. G, J. of Appl, Phys ., July i960. 

10., Smith, R. Ac Semiconductors , Cambridge University Press, 1959 - 

11. Loeb, L. Bo Kinetic Theory of Gases , Chap. 4. Dover: I96I0 

12 o Shockley, W. J. B. S. T, J. , Vol. 30, 1951. 

13 . Mott and Massey. Theory of Atomic Coll ., Oxford University Press , 1933 ° 

Ik. Conwell, Eo M. Proc. IRE 11 , 1952. 

15. Dingle, B. Phil. Mag ., Vol. 46, 1955= 

l6o Brookes, A„ Advances in Electronics and Electron Phys ., Vol. 8, 1955 » 

17. Chynoweth, A. Go J. of Appl. Phys ., July i960. 

18, Shockley, W. Solid- State Electronics , Vol. 2, No. 1, 1961. 

19 o Haitz ^ R„ Research Report , Shockley Lab, Clevite Co., Aug, 1963 ° 

20 o Tokuyama, T. Solid-State Electronics , Vol. 5, 1962 . 

21. Mclntyre, R. J. loc. cit. , page 986. 

22. Prince, M. B. Phys. Rev ., Vol. 8l, 1951. 

23° Do Ho Menzel. Fundamental Formulas of Physics . Dover: i960. 

2Uo Jonscher, A. K. Progress in Semicond o Ed., A. F. Gibson, Heywood and Co„, 

25. Valdes, L. Physical Theory of Transistors. McGraw-Hill, 1961. 


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